Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole. (English) Zbl 1302.35272
Summary: Let \(\mathbb A(\epsilon)\) be the annular domain obtained by removing from a bounded open domain \(\mathbb I^o\) of \(\mathbb R^n\) a small cavity of size \(\epsilon>0\). Then we assume that for some natural index \(l,\lambda_l[\mathbb I^o]>0\) is a simple Neumann eigenvalue of \(-\Delta\) in \(\mathbb I^o\), and we show that there exists a real valued real analytic function \(\hat\lambda_l(\cdot,\cdot)\) defined in an open neighborhood of \((0,0)\) in \(\mathbb R^2\) such that the \(l\)th Neumann eigenvalue \(\lambda_l[\mathbb A(\epsilon)]\) of \(-\Delta\) in \(\mathbb A(\epsilon)\) equals \(\hat\lambda_l(\epsilon,\kappa_n\epsilon\log\epsilon)\) and such that \(\hat\lambda_l(0,0)=\lambda_l[\mathbb I^o]\). Here, \(\kappa_n=1\) if \(n\) is even and \(\kappa_n=0\) if \(n\) is odd. Thus in particular, we show that if \(n\) is even \(\lambda_l[\mathbb A(\epsilon)]\) can be expanded into a convergent double series of powers of \(\epsilon\) and \(\epsilon\log\epsilon\) and that if \(n\) is odd \(\lambda_l[\mathbb A(\epsilon)]\) can be expanded into a convergent series of powers of \(\epsilon\). Then related statements have been proved for corresponding eigenfunctions.
MSC:
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 31B10 | Integral representations, integral operators, integral equations methods in higher dimensions |
| 45F15 | Systems of singular linear integral equations |
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
Keywords:
Neumann eigenvalues and eigenvectors; singularly perturbed domain; Laplace operator; real analytic continuation in Banach spaceReferences:
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